Critical curve for the weakly coupled system of damped wave equations with mixed nonlinearities
Dinh Van Duong, Tuan Anh Dao, Masahiro Ikeda
TL;DR
The paper analyzes a weakly coupled system of damped-wave equations with mixed nonlinearities $|v|^p$ and $|u_t|^q$ and establishes a sharp critical curve for $n=1,2$ given by $pq = 1 + rac{2}{n}$. Using harmonic-analysis techniques and diffusion-phenomenon insights, it derives global existence of small-data Sobolev solutions for $pq > 1 + rac{2}{n}$ in 1D and 2D, along with precise decay rates, and proves blow-up for $pq < 1 + rac{2}{n}$ via a test-function method, thereby showing how derivative-type nonlinearity shifts the critical threshold toward the origin. The work also discusses the limitations for $n o ext{higher}$ dimensions and outlines future directions including extensions to systems with derivative nonlinearities like $|v_t|^p$ and further harmonic-analysis tools. Overall, the results clarify the impact of time-derivative-type nonlinearities on the global behavior of damped-wave systems and provide a foundational threshold that informs both theory and potential applications in wave diffusion regimes.
Abstract
In this paper, we would like to consider the Cauchy problem for a weakly coupled system of semi-linear damped wave equations with mixed nonlinear terms. Our main objective is to draw conclusions about the critical curve of this problem using tools from Harmonic Analysis. Precisely, we obtain a new critical curve $pq = 1+ \frac{2}{n}$ for $n =1,2$ by proving global (in time) existence of small data Sobolev solutions when $pq > 1 +\frac{2}{n}$ and blow-up of weak solutions in finite time even for small data when $pq < 1+ \frac{2}{n}$ for $n \geq 1$. From this, we infer the impact of the nonlinearities of time derivative-type on the critical curve associated with the system.